LAS VEGAS — Declaring "now is the time" to fix broken immigration laws, President Barack Obama on Tuesday urged Congress to put millions of unauthorized immigrants on a clear path to U.S. citizenship while cracking down on businesses that employ people illegally and tightening security at the borders. He heralded a rare show of bipartisanship between the White House and Senate leaders on basic plans to resolve the long, emotional national issue.

But both the White House and Senate proposals for tackling the complex and emotionally charged issue still lack key details. And potential roadblocks are already emerging over how to structure the avenue to citizenship and whether a bill would cover same-sex couples — and that's all before a Senate measure can be debated, approved and sent to the Republican-controlled House where opposition is likely to be stronger.

Obama, in the heart of the heavily Latino Southwest, praised the Senate push, saying Congress is showing "a genuine desire to get this done soon." But mindful of previous immigrations efforts that have failed, Obama warned that the debate would become more difficult as it gets closer to a conclusion.

"The question now is simple," Obama said during a campaign-style event in Las Vegas, one week after being sworn in for a second term in the White House. "Do we have the resolve as a people, as a country, as a government to finally put this issue behind us? I believe that we do."

Despite possible obstacles to come, the broad agreement between the White House and bipartisan lawmakers in the Senate represents a drastic shift in Washington's willingness to tackle immigration, an issue that has languished for years. Much of that shift is politically motivated, due to the growing influence of Hispanics in presidential and other elections and their overwhelming support for Obama in November.

Still, some Republicans, including House Speaker John Boehner of Ohio and Senate Minority Leader of Kentucky, responded cautiously to the proposals from the president, on Tuesday, and the Senate group, which put forward its proposals one day earlier.

"Any solution should be a bipartisan one, and we hope the president is careful not to drag the debate to the left and ultimately disrupt the difficult work that is ahead in the House and Senate," said Brendan Buck, a Boehner spokesman.

The separate White House and Senate proposals focus on the same principles: providing a way for most of the estimated 11 million people already in the U.S. illegally to become citizens, strengthening border security, cracking down on employers who hire illegal immigrants and streamlining the legal immigration system.

A consensus around the question of citizenship could help lawmakers clear one major hurdle that has blocked previous immigration efforts. Many Republicans have opposed allowing illegal immigrants to become citizens, saying that would be an unfair reward for people who have broken the law.

Details on how to achieve a pathway to citizenship still could prove to be a major sticking point between the White House and the Senate group, which is comprised of eight lawmakers — four Democrats and four Republicans.